A Spectral Approach for Locally Assessing Time Series Model Misspecification

Tucker McElroy and Scott Holan

ABSTRACT

Peaks in the spectrum of a stationary process are indicative of
periodic phenomena, such as seasonality or business cycles. Hence
one important aspect of developing parametric models for periodic
processes is proper characterization of spectral peaks. By using
diagnostics constructed from the average ratio of two spectral
densities this work proposes to test whether a hypothesized model is locally
supported in the frequency domain by the data. The local fit of a
model is assessed by considering a subset of the whole frequency
band, focused on the locality of the spectral peak. This technique
can therefore be used to test the goodness-of-fit of a model with
respect to local frequency domain properties of the data. For
example, one can test for the appropriateness of a hypothesized
seasonal or cyclical spectral peak in the model for the data. In
the development of these diagnostics we provide a result of
independent interest, the asymptotic distribution of general
polynomial functionals of the periodogram. We present theoretical
properties for several new diagnostics, and also explore their
finite-sample properties through simulation and application to
several economic time series.

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